ELECTRONIC
COMMUNICATIONS in PROBABILITY
WHEN DOES A RANDOMLY WEIGHTED SELF–NORMALIZED
SUM CONVERGE IN DISTRIBUTION?
DAVID M. MASON 1
Statistics Program, University of Delaware, Newark, DE 19717 email: davidm@udel.edu
JOEL ZINN2
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368 email: email.com jzinn@math.tamu.edu
Submitted July 26, 2004, accepted in final form March 1, 2005 AMS 2000 Subject classification: 60F05
Keywords: Domain of attraction, self–normalized sums, regular variation
Abstract
We determine exactly when a certain randomly weighted self–normalized sum converges in distribution, partially verifying a 1965 conjecture of Leo Breiman, and then apply our results to characterize the asymptotic distribution of relative sums and to provide a short proof of a 1973 conjecture of Logan, Mallows, Rice and Shepp on the asymptotic distribution of self– normalized sums in the case of symmetry.
0.1
A conjecture of Breiman
Throughout this paper{Yi}i≥1 will denote a sequence of i.i.d. Y random variables, whereY is non–negative with distribution function G. Let Y ∈D(α), with 0< α≤2, denote thatY
is in the domain of attraction of a stable law of indexα. We shall use the notationY ∈D(0) to mean that 1−Gis a slowly varying function at infinity. Now let{Xi}i≥1 be a sequence of i.i.d. X random variables independent of{Yi}i≥1,whereX satisfies
E|X|<∞andEX = 0. (1) Consider the randomly weighted self–normalized sum
Rn =
Pn
i=1XiYi
Pn
i=1Yi
.
(Here and elsewhere we define 0/0 = 0.) In a beautiful paper, Breiman (1965) proved the following result characterizing whenRn converges in distribution to a non–degenerate law.
1RESEARCH PARTIALLY SUPPORTED BY NSA GRANT MDA904–02–1–0034 AND NSF GRANT DMS–0203865
2RESEARCH PARTIALLY SUPPORTED BY NSA GRANT H98230-04-1-0108
Theorem 1 Suppose for each such sequence{Xi}i≥1of i.i.dXrandom variables, independent of {Yi}i≥1, the ratioRn converges in distribution, and the limit law ofRn is non–degenerate
for at least one such sequence {Xi}i≥1. ThenY ∈D(α), with 0≤α <1.
Theorem 1 is a restatement of his Theorem 4. At the end of his 1965 paper Breiman conjectured that the conclusion of Theorem 1 remains true as long as there exist one i.i.d. X sequence {Xi}i≥1,satisfying (1), such thatRn converges in distribution to a non–degenerate law. We
shall provide a partial solution to his conjecture (we assumeE|X|p<∞for somep >2) and at the same time give a new characterization for a non-negative random variableY ∈D(α), with 0≤α <1.
Before we do this, let us briefly describe and comment upon Breiman’s proof of Theorem 1. Let
Dn(1)≥ · · · ≥Dn(n)≥0 (2)
denote the order values ofYj/Pni=1Yi, j= 1, . . . , n. Clearly along subsequences{n′} of{n},
the ordered random variablesD(ni), i= 1, . . . , n,converge in distribution to random sequences
{Di}i≥1 satisfyingDi≥0,i≥1, andP∞i=1Di= 1. From this one readily concludes that the
limit laws ofRn are of the form
∞
X
i=1
XiDi.
Breiman argues in his proof that if {D′
i}i≥1 is any other random sequence satisfyingDi′ ≥0, i≥1,P∞
i=1Di′ = 1 and
∞
X
i=1
XiDi′ =d
∞
X
i=1
XiDi, (3)
for all sequences {Xi}i≥1 of i.i.d. X random variables independent of{Yi}i≥1 satisfying (1), then
{D′
i}i≥1=d{Di}i≥1. (4) This implies that along the full sequence{n},
max 1≤j≤nYj/
n
X
i=1
Yi→dD1, (5)
whereD1 is either non–degenerate orD1 = 1. Breiman proves that whenD1= 1, Y ∈D(0), and whenD1 is non–degenerate necessarilyY ∈D(α) , with 0< α <1.
At first glance it may seem reasonable that it would be enough for (3) to hold for some i.i.d. X
sequence{Xi}i≥1 satisfying (1) in order to conclude (4). In fact, consider a sequence{si}i≥1 of independent Rademacher functions and let {ai}i≥1 and {bi}i≥1 be two sequences of non-increasing non-negative constants summing to 1. (By Rademacher we mean thatP{si= 1}= P{si=−1}= 1/2,for eachi≥1.) A special case of a result of Marcinkiewicz, see Theorem
5.1.5 in Ramachandran and Lau (1991), says that
∞
X
i=1
siai=d
∞
X
i=1
sibi
However, Jim Fill has shown that there exist two non–identically distributed random sequences {D′
i}i≥1and {Di}i≥1 such that
P∞
i=1siD′i =d P
∞
i=1siDi. Here is his example. Let{Di′}i≥1 equal to (1,0,0, . . . ,) with probability 1/5 and (1/4,1/4,1/4,1/4,0, . . .) with probability 4/5 and let{Di}i≥1 equal to (1/2,1/2,0, . . .) with probability 1/5 and (1/2,1/4,1/4,0, . . .) with probability 4/5. Clearly{D′
i}i≥1and{Di}i≥1are not equal in distribution. Whereas, calcula-tion verifies that (3) holds.
This indicates that one must look for another way to try to establish Breiman’s conjecture, than merely to refine his original proof. Our partial solution to Breiman’s conjecture is contained in the following theorem.
Theorem 2 Suppose that {Xi}i≥1 is a sequence of i.i.d. X random variables independent of {Yi}i≥1 , where X satisfies E|X|p < ∞ for some p >2 and EX = 0, then the ratio Rn
converges in distribution to a non-degenerate random variable R if and only if Y ∈D(α), with 0≤α <1.
The proof of Theorem 1 will follow readily from the following characterization of when Y ∈
D(α), with 0≤α <1. We shall soon see that whether Y ∈D(α), with 0≤α <1, or not depends on the limit ofET2
n,where
Tn:=
Pn
i=1siYi
Pn
i=1Yi
, (6)
with {si}i≥1 being a sequence of independent Rademacher random variables independent of {Yi}i≥1.
Proposition 3 We have Y ∈D(α), with 0≤α <1 if and only if
nE
µ Y
1
Pn
i=1Yi
¶2
→1−α. (7)
Remark 1 It can be inferred from Theorems 1, 2 and Proposition 1 of Breiman (1965) that the limit in (7) is equal to zero if and only if
max 1≤j≤nYj/
n
X
i=1
Yi→p0 (8)
if and only if there exist constantsBnրsuch that n
X
i=1
Yi/Bn→p1. (9)
Remark 2 Proposition 1 should be compared to a result of Bingham and Teugels (1981), which says
E
µ Pn
i=1Yi
max1≤i≤nYi
¶
→ρ, (10)
Proof. First assume thatY ∈D(α), where 0< α < 1. Notice that withTn defined as in
By Corollary 1 of Le Page, Woodroofe and Zinn (1981),
Tn →dT :=
We shall prove that
ET2=−Z
From this one gets from (13) after a little calculus that
ET2=α(1−α) Γ (1−α)Z
can write the expession in (14) as
which by the change of variables t=s/n1/α,
=
Z ∞
0
s
Z n
0
³
x−2/αexp³−sx−1/α´dx´
×
µ
1−1
n
Z n
0
h
1−exp³−sx−1/α´idx¶ n−1
ds.
A routine limit argument now shows that this last expression converges to −
Z ∞
0
sω′′(s) exp (−ω(s))ds.
Now assume thatET2
n →1−α,with 0< α <1. From equation (14) we get that ETn2=n
Z ∞
0
tϕ′′(t) (ϕ(t))n−1
dt→1−α,
where ϕ(t) = Eexp (−tY1), for t ≥ 0. Arguing as in the proof of Theorem 3 of Breiman (1965) this implies that
s
Z ∞
0
tϕ′′(t) exp (slogϕ(t))dt→1−α, ass→ ∞. (15)
For y ≥0, letq(y) denote the inverse of −logϕ(v). Changing variables to t= q(y) we get from (15) that
s
Z ∞
0
exp (−sy)q(y)ϕ′′(q(y))dq(y)→1−α, ass→ ∞.
By Karamata’s Tauberian theorem, see Theorem 1.7.1 on page 37 of Bingham et al (1987), we conclude that
v−1Z v 0
q(x)ϕ′′(q(x))dq(x)→1−α, asvց0,
which, in turn, by the change of variables y=q(x) gives
Rt
0yϕ
′′(y)dy
−logϕ(t) →1−α, as tց0. Since−log(1−s)/s→1 assց0,this implies that
Rt
0yϕ
′′(y)dy
1−ϕ(t) =
tϕ′(t)
1−ϕ(t)+ 1→1−α, as tց0, or in other words
tϕ′(t)
1−ϕ(t)→ −α, as tց0. (16)
Setf(x) =−x−2ϕ′(1/x) = (ϕ(1/x))′
, forx >0. With this notation we can rewrite (16) as
xf(x)
R∞
x f(y)dy
By Theorem 1.6.1 on page 30 of Bingham et al (1987) this implies that f(y) is regularly varying at infinity with index ρ =−α−1, which, in turn, by their Theorem 1.5.11 implies that 1−ϕ(1/x) is regularly varying at infinity with index −α, which says that 1−ϕ(s) is
which is regularly varying at 0 with indexα−1. Now by Theorem 1.7.1 on page 37 of Bingham et al (1987) this implies thatU(x) is regularly varying at infinity with index 1−α. This, in turn, by Theorem 1.7.2 on page 39 of Bingham et al (1987) implies that 1−G(x) is regularly varying at infinity with index−α. HenceG∈D(α).
To finish the proof we must show that
ETn2=nE
(Refer to Haeusler and Mason (1991) and the references therein.) Thus clearly whenever
Y ∈D(0) we have
Tn→d s1
and therefore we have (18). To go the other way, assume that (18) holds. This implies that
n
Proof of Theorem 2. First assume that for some non–degenerate random variableR,
Rn→dR. (20)
By Jensen’s inequality for anyr≥1,
This implies that wheneverRn→dR,whereRis non–degenerate, then
ER2n=EX2nE
µ Y
1
Pn
i=1Yi
¶2
→EX2(1−α),
where necessarily 0≤α <1. Thus by Proposition 1,Y ∈D(α),with 0≤α <1.
Breiman (1965) shows that whenever Y ∈D(α), with 0 ≤α <1, then (20) holds for some non–degenerate random variableR. To be specific, whenα= 0, R=dX and when 0< α <1,
it can be shown by using the methods of Le Page et al (1981) that
R=d
P∞
i=1Xi(Γi)−1/α
P∞
i=1(Γi)−1/α
. (21)
This completes the proof of Theorem 2. ⊔⊓
The proof just given is highly dependent on the assumption thatE|X|p<∞for some p >2.
To replace it by the weaker assumptionE|X|<∞would require an entirely different approach. Therefore the complete Breiman conjecture remains open. In the next section we provide some applications of our results to the study of the asymptotic distribution of relative ratio and self– normalized sums.
1
Applications
1.1
Application to relative ratio sums
Let {Yi}i≥1 be a sequence of i.i.d. Y non–negative random variables and for any n≥ 0 let
Sn=Pni=1Yi, whereS0:= 0. For anyn≥1 and 0≤t≤1,consider therelative ratio sum
Vn(t) := S[nt]
Sn
. (22)
Our first corollary characterizes when such relative ratio sums converge in distribution to a non–degenerate law.
Corollary 4 For any 0< t <1
Vn(t)→dV (t), (23)
where V(t) is non–degenerate if and only ifY ∈D(α), with 0≤α <1.
The proof Corollary 1 will be an easy consequence of the following proposition. Independent of {Yi}i≥1 let{ǫi(t)}i≥1 be a sequence of i.i.d. ǫ(t) random variables, whereP{ǫ(t) = 1}=
t= 1−P{ǫ(t) = 0},with 0< t <1. For any n≥1 and 0< t <1 let [nt] denote the integer part of ntand set
Nn(t) = n
X
i=1
ǫi(t).
Proposition 5 For all 0< t <1,
PNn(t)
i=1 Yi Sn
Proof of Proposition 2. We have
Proof of Corollary 1. Note that
PNn(t)
Therefore by Proposition 2 and (25), we readily conclude that (23) holds with a non–degenerate
V (t) if and only if
converges in distribution to a non-degenerate random variable. Thus Corollary 1 follows from Theorem 2. ⊔⊓
WhenY ∈D(0), it is easy to apply Proposition 2, (25) and (19) to get thatVn(t)→dǫ1(t), and whenY ∈D(α) , with 0< α <1, one gets from Proposition 2, (25) and by arguing as in Le Page et al (1981), that
Vn(t)→d
Also, one can show using Theorem 1, Theorem 2 and Proposition 1 of Breiman (1965) that
Pn
i=1(ǫi(t)−t)Yi Sn
→p0,
if and only if there exists a sequence of positive constantsBnրsuch that (9) holds.
Further-more, by Proposition 2 and (25), we see that this happens if and only ifVn(t)→pt.
An easy variation of Corollary 1, says that ifS′
n=dSn, withSn′ andSn independent, then techniques of Le Page et al (1981) one can show that
K=d W
′
where W′
α=dWα,Wα′ andWαare independent and Wα=d
∞
X
i=1 Γ−i1/α.
Curiously, it can be shown that
Lα:= logWα and logK
provide examples of random variables that have finite positive moments of any order, yet have distributions that are not uniquely determined by their moments. To see this, let hα denote
the density of Lα. Using known results about densities of stable laws that can be found in
Ibragimov and Linnik (1971) and Zolotarev (1986) it can be proved that Lα has all positive
moments and its densityhα is inC∞. Moreover, it is readily checked that
−
Z ∞
−∞
loghα(x)
1 +x2 dx <∞.
This implies that the distribution of Lα is not uniquely determined by its moments. Refer
to Lin (1997). Furthermore, by a result of Devinatz (1959), this in turn implies that the distribution of logK=dL′α−Lα,whereL′αis an independent copy ofL′α,is also not uniquely
determined by its moments.
1.2
Application to self–normalized sums
Let{Xi}i≥1be a sequence of i.i.d. X random variables and consider theself–normalized sums
Sn(2) =
Pn
i=1Xi
pPn i=1Xi2
.
Logan, Mallows, Rice and Shepp (1973) conjectured thatSn(2) converges in distribution to a
standard normal random variable if and only if EX = 0 and X ∈D(2),and more generally thatSn(2) converges in distribution to a non–degenerate random variable not concentrated on
two points if and only ifX ∈D(α), with 0< α≤2, whereEX = 0 if 0< α <1 andX is in the domain of attraction of a Cauchy law in the caseα= 1. The first conjecture was proved by Gin´e, G¨otze and Mason (1997) and the more general conjecture has been recently established by Chistyakov and G¨otze (2004). Griffin and Mason (1991) attribute to Roy Erickson an elegant proof of the first conjecture of Logan et al (1973) in the case when X is symmetric about 0. We shall use Proposition 1 to extend Erickson’s method to provide a short proof of the second conjecture of Logan et al (1973), for the symmetric about 0 case. In the following corollary s andY are independent random variables, whereP{s= 1} =P{s=−1} = 1/2. Since for a random variable X symmetric about 0, X =d sY,where Y =d |X|, it establishes
the second Logan et al (1973) conjecture in the symmetric case. It is also Corollary 1 of Chistyakov and G¨otze (2004). The proof of the second Logan et al (1973) conjecture without the simplifying assumption of symmetry is much more lengthy and requires a lot of serious analysis.
have
In the proof of Corollary 2 we describe the possible limit laws and when they occur. Proof of Corollary 2. WhensY ∈D(0), then by using (19) one readily gets that
Sn(2)→ds.
and when sY ∈D(2), Raikov’s theorem (see Lemma 3.2 in Gin´e et al (1997)), implies that for any non–decreasing positive sequence{ai}i≥1 such that
Pn
i=1siYi/an→d Z,whereZ is a
standard normal random variable, one hasPn
i=1Yi2/a2n→p1 , which gives Sn(2)→dZ.
Next assume thatSn(2)→d S(2),where S(2) is non–degenerate. By Khintchine’s inequality
1.3
A conjecture
As in Breiman (1965) we shall end our paper with a conjecture. For a sequence of i.i.d. positive random variables {Yi}i≥1, a sequence of independent Rademacher random variables {si}i≥1 independent of{Yi}i≥1 and 1≤p <2,we conjecture that
Sn(p) :=
Pn
i=1siYi
(Pn
i=1Y
p i )
1/p →dS(p), (27)
where S(p) is a non-degenerate random variable if and only if Y ∈D(α), where 0≤α < p. At present we can only verify it for casep= 1 and the limit casep= 2.
AcknowledgementsThe authors thank Jim Fill for permitting us to use his counterexample and Evarist Gin´e for many helpful discussions. Also they are grateful to the referee for pointing out a number of misprints and clarifications.
References
[1] N.H. Bingham, and J.L. Teugels. Conditions implying domains of attraction.Proceedings of the Sixth Conference on Probability Theory (Bracsov, 1979), (1981) pp. 23–34, Ed. Acad. R. S. Romˆania, Bucharest.
[2] N.H. Bingham , C. M. Goldie and J. L. Teugels. Regular Variation. Encyclopedia of Mathematics and its Applications, (1987) 27, Cambridge University Press, Cambridge. [3] L. Breiman, On some limit theorems similar to the arc-sin law.Teor. Verojatnost. i
Prime-nen. (1965)10, pp. 351–360.
[4] G. P. Chistyakov and F. G¨otze Limit distributions of studentized sums. Ann. Probab., (2004) 32, 28–77.
[5] D. A. Darling. The influence of the maximum term in the addition of independent random variables.Trans. Amer. Math. Soc.(1952)73, 95-107.
[6] A. Devinatz. On a theorem of L´evy-Raikov.Ann. Math. Statist(1959)30, 583–586. [7] E. Gin´e, F. G¨otze and D. M. Mason. When is the Student t-statistic asymptotically
standard normal? Ann. Probab.(1997)251514–1531.
[8] P. S. Griffin and , D. M. Mason. On the asymptotic normality of self–normalized sums. Proc. Cambridge Phil. Soc.(1991) 109597-610.
[10] I. A. Ibragimov and Yu. V. Linnik. Independent and Stationary Sequences of Random Variables. With a Supplementary Chapter by I. A. Ibragimov and V. V. Petrov. Transla-tion from the Russian edited by J. F. C. Kingman (1971) Wolters-Noordhoff Publishing, Groningen.
[11] R. LePage, M. Woodroofe and J. Zinn. Convergence to a stable distribution via order statistics.Ann. Probab (1981)9713–752.
[12] G. D. Lin. On the moment problems.Statist. Probab. Lett(1997)35 85–90.
[13] B. F. Logan, C. L. Mallows, S. O. Rice and L. Shepp. Limit distributions of self-normalized sums.Ann. Probab.(1973) 1788–809.
[14] B. Ramachandran and K. Lau. Functional Equations in Probability Theory. Academic Press, Inc., Boston, MA (1991).
